{
  "id": "1904.07716",
  "version": "v1",
  "published": "2019-04-15T16:41:27.000Z",
  "updated": "2019-04-15T16:41:27.000Z",
  "title": "On the Hilbert scheme of linearly normal curves in $\\mathbb{P}^r$ of relatively high degree",
  "authors": [
    "Edoardo Ballico",
    "Claudio Fontanari",
    "Changho Keem"
  ],
  "comment": "10 pages. arXiv admin note: substantial text overlap with arXiv:1903.02307",
  "categories": [
    "math.AG"
  ],
  "abstract": "We denote by $\\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\\mathbb{P}^r$. We let $\\mathcal{H}^\\mathcal{L}_{d,g,r}$ be the union of those components of $\\mathcal{H}_{d,g,r}$ whose general element is linearly normal. In this article, we show that any non-empty $\\mathcal{H}^\\mathcal{L}_{d,g,r}$ ($d\\ge g+r-3$) is irreducible in an extensive range of the triple $(d,g,r)$ beyond the Brill-Noether range. This establishes the validity of a certain modified version of an assertion of Severi regarding the irreducibility of the Hilbert scheme $\\mathcal{H}^\\mathcal{L}_{d,g,r}$ of linearly normal curves for $g+r-3\\le d\\le g+r$, $r\\ge 5$, and $g \\ge 2r+3$ if $d=g+r-3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-15T16:41:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C05",
      "14H10"
    ],
    "keywords": [
      "linearly normal curves",
      "hilbert scheme",
      "relatively high degree",
      "general point corresponds",
      "smooth curves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}